, Volume 76, Issue 4, pp 1245–1263 | Cite as

An FPTAS for the Volume Computation of 0-1 Knapsack Polytopes Based on Approximate Convolution

  • Ei AndoEmail author
  • Shuji Kijima


Computing high dimensional volumes is a hard problem, even for approximation. Several randomized approximation techniques for #P-hard problems have been developed in the three decades, while some deterministic approximation algorithms are recently developed only for a few #P-hard problems. Motivated by a new technique for a deterministic approximation, this paper is concerned with the volume computation of 0-1 knapsack polytopes, which is known to be #P-hard. This paper presents a new technique based on approximate convolutions for a deterministic approximation of volume computations, and provides a fully polynomial-time approximation scheme for the volume computation of 0-1 knapsack polytopes. We also give an extension of the result to multi-constrained knapsack polytopes with a constant number of constraints.


Approximate convolution Volume computation #P-hard  Knapsack polytope 



The authors would like to thank the anonymous reviewers for their valuable comments. This work is partly supported by Grant-in-Aid for Scientific Research on Innovative Areas MEXT Japan “Exploring the Limits of Computation (ELC)” (Nos. 24106008, 24106005).


  1. 1.
    Ando, E., Kijima, S.: An FPTAS for the volume computation of 0-1 knapsack polytopes based on approximate convolution integral. Lect. Notes Comput. Sci. 8889, 376–386 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bandyopadhyay, A., Gamarnik, D.: Counting without sampling: asymptotics of the log-partition function for certain statistical physics models. Random Struct. Algorithms 33, 452–479 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bárány, I., Füredi, Z.: Computing the volume is difficult. Discrete Comput. Geom. 2, 319–326 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bayati, M., Gamarnik, D., Katz, D., Nair, C., Tetali, P.: Simple deterministic approximation algorithms for counting matchings. In: Proceedings of STOC, pp. 122–127 (2007)Google Scholar
  5. 5.
    Cousins, B., Vempala, S., Bypassing, K.L.S.: Gaussian Cooling and an \(O^\ast (n^3)\) Volume Algorithm. In: Proceedings of STOC, pp. 539–548 (2015)Google Scholar
  6. 6.
    Dyer, M.: Approximate counting by dynamic programming. In: Proceedings of STOC, pp. 693–699 (2003)Google Scholar
  7. 7.
    Dyer, M., Frieze, A.: On the complexity of computing the volume of a polyhedron. SIAM J. Comput. 17(5), 967–974 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dyer, M., Frieze, A., Kannan, R.: A random polynomial-time algorithm for approximating the volume of convex bodies. J. Assoc. Comput. Mach. 38(1), 1–17 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Elekes, G.: A geometric inequality and the complexity of computing volume. Discrete Comput. Geom. 1, 289–292 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gamarnik, D., Katz, D.: Correlation decay and deterministic FPTAS for counting colorings of a graph. J. Discrete Algorithms 12, 29–47 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gopalan, P., Klivans, A., Meka, R.: Polynomial-time approximation schemes for knapsack and related counting problems using branching programs. arXiv:1008.3187v1 (2010)
  12. 12.
    Gopalan, P., Klivans, A., Meka, R., Štefankovič, D., Vempala, S., Vigoda, E.: An FPTAS for #knapsack and related counting problems. In: Proceedings of FOCS, pp. 817–826 (2011)Google Scholar
  13. 13.
    Li, L., Lu, P., Yin, Y.: Approximate counting via correlation decay in spin systems. In: Proceedings of SODA, pp. 922–940 (2012)Google Scholar
  14. 14.
    Li, L., Lu, P., Yin, Y.: Correlation decay up to uniqueness in spin systems. In: Proceedings of SODA, pp. 67–84 (2013)Google Scholar
  15. 15.
    Li, J., Shi, T.: A fully polynomial-time approximation scheme for approximating a sum of random variables. Oper. Res. Lett. 42, 197–202 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lin, C., Liu, J., Lu, P.: A simple FPTAS for counting edge covers. In: Proceedings of SODA, pp. 341–348 (2014)Google Scholar
  17. 17.
    Lovász, L.: An algorithmic theory of numbers, graphs and convexity. SIAM Society for Industrial and Applied Mathematics, Philadelphia (1986)CrossRefzbMATHGoogle Scholar
  18. 18.
    Lovász, L., Vempala, S.: Simulated annealing in convex bodies and an \(O^\ast (n^4)\) volume algorithm. J. Comput. Syst. Sci. 72, 392–417 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Štefankovič, D., Vempala, S., Vigoda, E.: A deterministic polynomial-time approximation scheme for counting knapsack solutions. SIAM J. Comput. 41(2), 356–366 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Weitz, D.: Counting independent sets up to the tree threshold. In: Proceedings of STOC, pp. 140–149 (2006)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Sojo UniversityKumamotoJapan
  2. 2.Kyushu UniversityFukuokaJapan

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